`((1)/(3)*(1)/(3)*(1)/(3)+(1)/(4)*(1)/(4)*(1)/(4)-3.(1)/(3)*(1)/(4)*(1)/(5)+(1)/(5)*(1)/(5)*(1)/(5))/((1)/(3)*(1)/(3)+(1)/(4)*(1)/(4)+(1)/(5)*(1)/(5)-((1)/(3)*(1)/(4)+(1)/(4)*(1)/(5)+(1)/(5)*(1)/(3)))` is equal to

Options

A. `(2)/(3)`
B. `(3)/(4)`
C. `(47)/(60)`
D. `(49)/(60)`

Answer

To solve the expression

\[ \frac{\left(\frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3} + \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4} - 3 \cdot \frac{1}{3} \cdot \frac{1}{4} \cdot \frac{1}{5} + \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}\right)}{\left(\frac{1}{3} \cdot \frac{1}{3} + \frac{1}{4} \cdot \frac{1}{4} + \frac{1}{5} \cdot \frac{1}{5} - \left(\frac{1}{3} \cdot \frac{1}{4} + \frac{1}{4} \cdot \frac{1}{5} + \frac{1}{5} \cdot \frac{1}{3}\right)}\right) \]

we can simplify it step by step.

### Step 1: Define Variables
Let:
- \( A = \frac{1}{3} \)
- \( B = \frac{1}{4} \)
- \( C = \frac{1}{5} \)

### Step 2: Calculate the Numerator
The numerator is:
\[ A^3 + B^3 - 3ABC + C^3 \]

Calculating each term:
- \( A^3 = \left(\frac{1}{3}\right)^3 = \frac{1}{27} \)
- \( B^3 = \left(\frac{1}{4}\right)^3 = \frac{1}{64} \)
- \( C^3 = \left(\frac{1}{5}\right)^3 = \frac{1}{125} \)
- \( 3ABC = 3 \cdot \frac{1}{3} \cdot \frac{1}{4} \cdot \frac{1}{5} = 3 \cdot \frac{1}{60} = \frac{3}{60} = \frac{1}{20} \)

Now, substituting these values into the numerator:
\[ \frac{1}{27} + \frac{1}{64} - \frac{1}{20} + \frac{1}{125} \]

### Step 3: Find a Common Denominator
The least common multiple (LCM) of 27, 64, 20, and 125 is 10800. We convert each fraction:
- \( \frac{1}{27} = \frac{400}{10800} \)
- \( \frac{1}{64} = \frac{168.75}{10800} \) (or \( \frac{675}{10800} \))
- \( \frac{1}{20} = \frac{540}{10800} \)
- \( \frac{1}{125} = \frac{86.4}{10800} \) (or \( \frac{86.4}{10800} \))

Now, we can add and subtract these fractions:
\[ \frac{400 + 675 - 540 + 86.4}{10800} \]

### Step 4: Calculate the Numerator Value
Calculating:
\[ 400 + 675 - 540 + 86.4 = 621.4 \]
Thus, the numerator becomes:
\[ \frac{621.4}{10800} \]

### Step 5: Calculate the Denominator
The denominator is:
\[ A^2 + B^2 + C^2 - (AB + BC + CA) \]

Calculating each term:
- \( A^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \)
- \( B^2 = \left(\frac{1}{4}\right)^2 = \frac{1}{16} \)
- \( C^2 = \left(\frac{1}{5}\right)^2 = \frac{1}{25} \)

Now, \( AB + BC + CA \):
- \( AB = \frac{1}{3} \cdot \frac{1}{4} = \frac{1}{12} \)
- \( BC = \frac{1}{4} \cdot \frac{1}{5} = \frac{1}{20} \)
- \( CA = \frac{1}{5} \cdot \frac{1}{3} = \frac{1}{15} \)

Now, substituting these values into the denominator:
\[ \frac{1}{9} + \frac{1}{16} + \frac{1}{25} - \left(\frac{1}{12} + \frac{1}{20} + \frac{1}{15}\right) \]

### Step 6: Find a Common Denominator for the Denominator
The LCM of 9, 16, 25, 12, 20, and 15 is 1800.

Calculating each term:
- \( \frac{1}{9} = \frac{200}{1800} \)
- \( \frac{1}{16} = \frac{112.5}{1800} \) (or \( \frac{112.5}{1800} \))
- \( \frac{1}{25} = \frac{72}{1800} \)
- \( \frac{1}{12} = \frac{150}{1800} \)
- \( \frac{1}{20} = \frac{90}{1800} \)
- \( \frac{1}{15} = \frac{120}{1800} \)

Now, we can add and subtract these fractions:
\[ \frac{200 + 112.5 + 72 - (150 + 90 + 120)}{1800} \]

### Step 7: Calculate the Denominator Value
Calculating:
\[ 200 + 112.5 + 72 - 360 = 24.5 \]
Thus, the denominator becomes:
\[ \frac{24.5}{1800} \]

### Step 8: Final Calculation
Now we have:
\[ \frac{\frac{621.4}{10800}}{\frac{24.5}{1800}} = \frac{621.4 \cdot 1800}{10800 \cdot 24.5} \]

### Step 9: Simplify
This simplifies to:
\[ \frac{621.4 \cdot 1800}{10800 \cdot 24.5} = \frac{621.4 \cdot 1}{60 \cdot 24.5} = \frac{621.4}{1470} \]

### Step 10: Final Result
Calculating this gives us the final value.